Finite Group Databasebeta

← #26133200

Subgroups — #26133200

Order 12032.

466 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256490C211yesmaximal, characteristic, complemented
2#20252107C211yesmaximal, characteristic
3#20245017C211yesmaximal, characteristic, complemented
4#20245015C211yesmaximal, characteristic, complemented
5#20245008C211yesmaximal, characteristic, complemented
6#20241468C211yesmaximal, characteristic, complemented
7C47×#1375C211yesmaximal, characteristic, complemented
8C2×#13094030C2211yescharacteristic
9C2×#13094030C2211yescharacteristic
10#15689673C411yescharacteristic
11#15689541—22no—
12#15689539—22no—
13C2×#13093964C2211yescharacteristic
14C2×#13093964C2211yescharacteristic
15M32×D47—22no—
16#15689196—22no—
17#15689196—22no—
18#15689147C411yescharacteristic
19#15689089C411yescharacteristic
20#15689089C411yescharacteristic
21#15688908—22no—
22#15688906—22no—
23#15688904—22no—
24C2×M32×C47C2211yescharacteristic
25C2×M32×C47C2211yescharacteristic
26C2×#97×C47C2211yescharacteristic
27C2×#10625350C2×C411yescharacteristic
28C2×#10625350C2×C411yescharacteristic
29#13094030—21no—
30#13094030—21no—
31#13094030—21no—
32#13094030—21no—
33#13094030—21no—
34#13094030—21no—
35#13094030—21no—
36#13094030—21no—
37C2×#10625329C2×C411yescharacteristic
38C2×#10625329C2×C411yescharacteristic
39C2×#10625324—21no—
40C2×#10625324D412yes—
41C2×C8×D47—21no—
42C2×C8×D47D412yes—
43#13094024C811yescharacteristic
44#13094021C811yescharacteristic
45#13094012C811yescharacteristic
46#13094012C811yescharacteristic
47#13093998—22no—
48C4×D188—22no—
49#13093964—21no—
50#13093964—21no—
51#13093964—21no—
52#13093964—21no—
53#13093963—24no—
54C2×C47⋊C16—21no—
55C2×C47⋊C16D412yescomplemented
56C16×D47—24no—
57C4⋊(C47⋊C8)—22no—
58(C2×D47)⋊C8—22no—
59(C2×D47)⋊C8—22no—
60C22×C8×C47C2×C411yescharacteristic
61C22×C8×C47C2311yescharacteristic
62C2×C42×C47C2×C411yescharacteristic
63M32×C47—21no—
64M32×C47—21no—
65M32×C47—21no—
66M32×C47—21no—
67C2×C16×C47—21no—
68C2×C16×C47D412yescomplemented
69#97×C47—21no—
70#97×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47C22×C411yescharacteristic
73C2×C22⋊D47—21no—
74C2×C22⋊D47M1612yes—
75#10625350—21no—
76#10625350—21no—
77#10625350—21no—
78#10625350—21no—
79#10625350—21no—
80#10625350—21no—
81#10625350—21no—
82#10625350—21no—
83C2×D188—22no—
84C2×D188C2×C811yescharacteristic
85C2×D188C2×C811yescharacteristic
86C2×C4×D47—21no—
87C2×C4×D47C22⋊C412yes—
88C2×Dic94C2×C811yescharacteristic
89C2×Dic94C2×C811yescharacteristic
90C2×C8×C47—21no—
91C2×C8×C47—21no—
92C2×C8×C47—21no—
93C2×C8×C47C2×D411yescharacteristic
94C2×C8×C47C2×D411yescharacteristic
95C2×C8×C47C2×D411yescharacteristic
96C2×C8×C47C2×D411yescharacteristic
97C2×C8×C47#4111yescharacteristic
98C2×C8×C47#4111yescharacteristic
99C42×C47—21no—
100C42×C47C2×C811yescharacteristic
101C42×C47C2×C811yescharacteristic
102(C2×D47)⋊C4—22no—
103(C2×D47)⋊C4—22no—
104C4⋊Dic47—22no—
105#10625329—21no—
106#10625329—21no—
107#10625329—21no—
108#10625329—21no—
109C2×C47⋊C8—21no—
110C2×C47⋊C8C22⋊C412yes—
111#10625324—22no—
112#10625324—22no—
113#10625324—41no—
114#10625324—41no—
115C8×D47—22no—
116C8×D47—22no—
117C8×D47—41no—
118C8×D47—41no—
119C16×C47—22no—
120C16×C47—41no—
121C47⋊C16—22no—
122C47⋊C16—41no—
123C23×C47C2×M1611yescharacteristic
124C22×D47—21no—
125C22×D47C22⋊C812yes—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—81no—
129C2×C4×C47C2×C22⋊C411yescharacteristic
130C2×C4×C47C4×D411yescharacteristic
131C2×C4×C47C4×D411yescharacteristic
132C2×C4×C47C22×C811yescharacteristic
133C2×C4×C47#13111yescharacteristic
134C2×C4×C47#13111yescharacteristic
135C22⋊D47—22no—
136C22⋊D47—22no—
137C22⋊D47—41no—
138C22⋊D47—41no—
139C2×Dic47—21no—
140C2×Dic47C22⋊C812yes—
141D188—21no—
142D188—21no—
143D188—21no—
144D188—21no—
145D188—82no—
146C4×D47—22no—
147C4×D47—22no—
148C4×D47—41no—
149C4×D47—41no—
150Dic94—21no—
151Dic94—21no—
152Dic94—21no—
153Dic94—21no—
154C8×C47—41no—
155C8×C47#12011yescharacteristic
156C8×C47#12111yescharacteristic
157C8×C47#12111yescharacteristic
158C8×C47#12311yescharacteristic
159C47⋊C8—22no—
160C47⋊C8—41no—
161#12988—471nomaximal, Sylow
162C22×C47—41no—
163C22×C47C2×C22⋊C811yescharacteristic
164C22×C47#43511yescharacteristic
165C22×C47#43511yescharacteristic
166C2×D47—22no—
167C2×D47—42no—
168C2×D47—42no—
169C2×D47—41no—
170C4×C47—41no—
171C4×C47—41no—
172C4×C47#38611yescharacteristic
173C4×C47#40811yescharacteristic
174C4×C47C8×D411yescharacteristic
175C4×C47C8×D411yescharacteristic
176Dic47—22no—
177Dic47—41no—
178#3407—471no—
179#2932—471no—
180#2117—471no—
181#2117—471no—
182#2115—471no—
183#1375—471no—
184#1375D4711yescharacteristic, complemented
185C2×C47—21no—
186C2×C47—21no—
187C2×C47—21no—
188C2×C47#186911yescharacteristic
189D47—42no—
190D47—81no—
191C2×#131—471no—
192C2×#131—471no—
193#532—471no—
194#504—942no—
195#504—942no—
196C2×M32—471no—
197C2×M32—471no—
198C2×M32—942no—
199C2×M32C2×D4711yescharacteristic
200C2×M32C2×D4711yescharacteristic
201#435—942no—
202#435—942no—
203#423—471no—
204#408—471no—
205#408—471no—
206C2×#97C2×D4711yescharacteristic
207#350—942no—
208#350—942no—
209#349—942no—
210C47#1298811yescharacteristic, complemented, Sylow
211C2×#41—471no—
212C2×#41—471no—
213#131—941no—
214#131—941no—
215#131—941no—
216#131—941no—
217#131—941no—
218#131—941no—
219#131—941no—
220#131—941no—
221C2×M16—471no—
222C2×M16—472no—
223C2×M16—471no—
224C2×M16—941no—
225C22×C8—472no—
226C22×C8—941no—
227C22×C8C4×D4711yescharacteristic
228C22×C8C22×D4711yescharacteristic
229C4⋊Q8—471no—
230#127—471no—
231#124—471no—
232#124—471no—
233#121—942no—
234C4×D4—942no—
235C2×C42C4×D4711yescharacteristic
236M32—21no—
237M32—21no—
238M32—21no—
239M32—21no—
240M32—944no—
241M32—941no—
242M32—941no—
243M32—941no—
244M32—941no—
245C2×C16—21no—
246C2×C16—472no—
247C2×C16—944no—
248C2×C16—941no—
249C2×C16D18812yescomplemented
250C4⋊C8—942no—
251C22⋊C8—942no—
252C22⋊C8—942no—
253#97—21no—
254#97—21no—
255#41—941no—
256#41—941no—
257#41—941no—
258#41—941no—
259#41—941no—
260#41—941no—
261#41—941no—
262#41—941no—
263C2×Q8—471no—
264C2×Q8—471no—
265C2×D4—472no—
266C2×D4—471no—
267C2×D4—471no—
268C2×D4—942no—
269C2×D4—941no—
270C22×C4—21no—
271C22×C4—472no—
272C22×C4—941no—
273C22×C4C2×C4×D4711yescharacteristic
274M16—941no—
275M16—941no—
276M16—942no—
277M16—942no—
278M16—941no—
279M16—941no—
280M16—1881no—
281M16—1881no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—472no—
286C2×C8—942no—
287C2×C8—942no—
288C2×C8—941no—
289C2×C8—1881no—
290C2×C8—1881no—
291C2×C8C2×D18811yescharacteristic
292C2×C8#1062535011yescharacteristic
293C2×C8D4×D4711yescharacteristic
294C2×C8D4×D4711yescharacteristic
295C2×C8D4×D4711yescharacteristic
296C2×C8#1062535411yescharacteristic
297C4⋊C4—942no—
298C22⋊C4—942no—
299C22⋊C4—942no—
300C42—21no—
301C42C8×D4711yescharacteristic
302C42C8×D4711yescharacteristic
303C16—22no—
304C16—41no—
305C16—942no—
306C16—1881no—
307C23—472no—
308C23—941no—
309C23M16×D4711yescharacteristic
310Q8—941no—
311Q8—941no—
312Q8—941no—
313Q8—941no—
314D4—942no—
315D4—942no—
316D4—941no—
317D4—941no—
318D4—941no—
319D4—941no—
320D4—3762no—
321D4—1881no—
322D4—1881no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—81no—
326C2×C4—472no—
327C2×C4—942no—
328C2×C4—942no—
329C2×C4—941no—
330C2×C4—1881no—
331C2×C4—1881no—
332C2×C4C22⋊C4×D4711yescharacteristic
333C2×C4#1309398611yescharacteristic
334C2×C4C4×D18811yescharacteristic
335C2×C4C2×C8×D4711yescharacteristic
336C2×C4#1309403011yescharacteristic
337C2×C4#1309403211yescharacteristic
338C8—41no—
339C8—942no—
340C8—1881no—
341C8#1309399411yescharacteristic
342C8#1309399711yescharacteristic
343C8#1309399911yescharacteristic
344C8#1309400911yescharacteristic
345C22—41no—
346C22—942no—
347C22—1882no—
348C22—1882no—
349C22—1881no—
350C22C22⋊C8×D4711yescharacteristic
351C22#1568919311yescharacteristic
352C22#1568919611yescharacteristic
353C4—41no—
354C4—41no—
355C4—942no—
356C4—1881no—
357C4#1568900111yescharacteristic
358C4#1568908811yescharacteristic
359C4#1568918811yescharacteristic
360C4C8×D18811yescharacteristic
361C2—21no—
362C2—21no—
363C2—21no—
364C2—3761no—
365C2—1882no—
366C2#2024367911yescharacteristic
subgroupG/H
centerC4#15689001
commutatorC2×C4×C47C22×C8
FrattiniC22×C8C22×D47
FittingC47×#1375C2
socleC2×C47#1869
prime radical#15689673C4
cosocleC22×C8×C47C23
hypercenter#1375D47
hypocenterC47#12988
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C22×C8 ◁ #1375

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #20256490C2semidirect
3. #20245017C2semidirect
4. #20245015C2semidirect
5. #20245008C2semidirect
6. #20241468C2semidirect
7. C47×#1375C2semidirect
55. C2×C47⋊C16D4semidirect
68. C2×C16×C47D4semidirect
184. #1375D47semidirect
210. C47#12988semidirect
249. C2×C16D188semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202564902
#202521072
#202450172
#202450152
#202450082
#202414682
C47×#13752
#1298847